Bootstrapping ${\mathcal N}=2$ chiral correlators

TL;DR

The paper develops a numerical bootstrap framework for four-point functions of chiral operators in 4d ${ m N}=2$ SCFTs, extending from identical to mixed-dimension correlators to probe Coulomb-branch physics and the Zamolodchikov metric. By constructing and solving crossing equations with superconformal blocks, it yields bounds on OPE coefficients and operator dimensions, and applies the method to the rank-one Argyres-Douglas theory $H_0$, suggesting it sits at a kink consistent with a vanishing of a specific short multiplet and a minimal central charge ${c= frac{11}{30}}$. In the mixed-correlator setup, the authors constrain central charges, Coulomb-branch relations, and curvature components of the Zamolodchikov metric, demonstrating that mixed data refine the allowed theory space and can rule out certain chiral-ring relations. Overall, the work demonstrates that the N=2 bootstrap, especially with mixed correlators, can carve out islands in the space of ${ m N}=2$ SCFTs and provide quantitative handles on non-Lagrangian theories. Future directions include extending to more multiplets, improving blocks, and seeking islands or unique solutions for other rank theories.

Abstract

We apply the numerical bootstrap program to chiral operators in four-dimensional ${\mathcal N}=2$ SCFTs. In the first part of this work we study four-point functions in which all fields have the same conformal dimension. We give special emphasis to bootstrapping a specific theory: the simplest Argyres-Douglas fixed point with no flavor symmetry. In the second part we generalize our setup and consider correlators of fields with unequal dimension. This is an example of a mixed correlator and allows us to probe new regions in the parameter space of ${\mathcal N}=2$ SCFTs. In particular, our results put constraints on relations in the Coulomb branch chiral ring and on the curvature of the Zamolodchikov metric.

Figures (13)

• Figure 1: Upper and lower bound on ${\mathcal{C}}_{0,2r_1- 1 (0,1)}$ for $\Lambda=12,16,20$. The panel on the right zooms in the region around $r_1=\tfrac{6}{5}$ in order to single out the $H_0$ family.
• Figure 2: Bound on the dimension of the first unprotected scalar in the chiral channel for $\Lambda=12,16,20$. The solid (dashed) line allows (disallows) the short ${\mathcal{B}}_{1,2r_1-1 (0,0)}$ multiplet. The dimensions shown correspond to the superconformal descendant that makes an appearance in the OPE. The black dots mark the external dimension corresponding to known theories listed in Tabs. \ref{['Tab:rank1theories']} and \ref{['Tab:newrank1theories']}, and they are drawn on the bounds allowing or disallowing for the short multiplet according to whether or not these theories have a mixed branch.
• Figure 3: Left: Upper bound on the OPE coefficient of the ${\mathcal{B}}_{R=1, r=2r_1-1 (0,0)}$ multiplet, for external dimension $r_1=\tfrac{6}{5}$, as a function of the central charge at $\Lambda=12$. The vertical dashed line corresponds to the minimum central charge allowed numerically with $\Lambda=12$. Right: Minimum allowed central charge for varying $\Lambda$, the dashed horizontal line marks the central charge of the rank one $H_0$ theory. The middle orange line shows a linear fit to all the data points, while the top and bottom blue lines show fits to different subsets of the points.
• Figure 4: Left: Bound on the dimension of the second long scalar $\Delta_0^\prime$ in the $\phi \phi$ channel as a function of the dimension of the first long scalar $\Delta_0$, for external dimension $r_1=\tfrac{6}{5}$. The central charge is left arbitrary and $\Lambda=14,16,18,20$. The black curve corresponds to the region where the short multiplet is required to have a positive OPE coefficient. The shaded black region is always excluded, and the shaded blue region only if one demands the absence of the ${\mathcal{B}}_{1,2r_1-1 (0,0)}$ short multiplet. Right: Upper and lower bounds on the OPE coefficient of the ${\mathcal{B}}_{1,2r_1-1 (0,0)}$ multiplet, as a function of the dimension of the first long scalar $\Delta_0$.
• Figure 5: Bound on the dimension of the second long scalar $\Delta_0^\prime$ in the $\phi_{r_1} \times \phi_{r_1}$ channel as a function of the dimension of the first long scalar $\Delta_0$, for external dimension $r_1=\tfrac{6}{5}$ with $\Lambda=16$. The different colors correspond to different central charges.